A bigger question is how much conversion would do for a Jew. Quite a lot pre 1890s, but probably much less 1910-1970.
Barry Goldwater comes close to counting as an example of this — he wasn’t a
convert, as he was raised Episcopalian, but he was ethnically half-Jewish through his father. Obviously, to say the least, he had a lot of issues which negatively affected his electability, but I don’t think that this was one of them; after all, the handful of states he did win were probably the most anti-Semitic in the country. I suppose one can question how widely this fact about his ancestry was actually known, but his surname is fairly clearly an anglicised German Jewish one.